What is the calcium equivalent role in neural networks
Neural networks don't model biological neurons. 

They are at best inspired by biological neurons, in that they get excited by certain inputs and fire once the excitation crosses a threshold. And this second point even holds only approximately because the backpropagation algorithm needs smoothed out steps to learn by gradient descent. And backpropagation is not even inspired by biology, that would rather be hebbian learning. 

Generally in machine learning people do what works. The connection to biology is tenuous at best. You will usually not find one to one correspondence of low level details between machine learning setups and biological neurons. For that you'll have to turn to brain simulations.